ECCC-Report TR20-069https://eccc.weizmann.ac.il/report/2020/069Comments and Revisions published for TR20-069en-usMon, 04 May 2020 01:01:57 +0300
Paper TR20-069
| Optimal Error Pseudodistributions for Read-Once Branching Programs |
Eshan Chattopadhyay,
Jyun-Jie Liao
https://eccc.weizmann.ac.il/report/2020/069In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length $n$ and width $w$ read-once branching programs with seed length $O(\log n\cdot \log(nw)+\log n\cdot\log(1/\varepsilon))$ and error $\varepsilon$. It remains a central question to reduce the seed length to $O(\log (nw/\varepsilon))$, which would prove that $\mathbf{BPL}=\mathbf{L}$. However, there has been no improvement on Nisan's construction for the case $n=w$, which is most relevant to space-bounded derandomization.
Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced the notion of a \emph{pseudorandom pseudo-distribution} (PRPD) and gave an explicit construction of a PRPD with seed length $\tilde{O}(\log n\cdot \log(nw)+\log(1/\varepsilon))$. A PRPD is a relaxation of a pseudorandom generator, which suffices for derandomizing $\mathbf{BPL}$ and also implies a hitting set. Unfortunately, their construction is quite involved and complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler hitting set generator with seed length $O(\log n\cdot \log(nw)+\log(1/\varepsilon))$, but their techniques are restricted to hitting sets.
In this work, we construct a PRPD with seed length
$$O(\log n\cdot \log (nw)\cdot \log\log(nw)+\log(1/\varepsilon)).$$
This improves upon the construction in \cite{BCG18} by a $O(\log\log(1/\varepsilon))$ factor, and is optimal in the small error regime. In addition, we believe our construction and analysis to be simpler than the work of Braverman, Cohen and Garg.Mon, 04 May 2020 01:01:57 +0300https://eccc.weizmann.ac.il/report/2020/069